싶어지다 means to start to want something. In general, -어지다 after an adjective (형용사) means to change the state to be (something).
물이 뜨겁다. The water is hot.
물이 뜨거워졌다. The water got hot.
물이 뜨거워진다. The water is getting hot.
배가 고프다. I'm hungry
라면을 보니 배가 고파졌다. I got hungry at the sight of ramen.
라면(이/을) 먹고 싶다. I want to eat ramen.
라면(이/을) 먹고 싶었다. I wanted to eat ramen. (I probably don't now.)
라면(이/을) 먹고 싶어졌다. ~ I didn't, but now I want to eat ramen.
라면(이/을) 먹고 싶어졌었다. (rare) ~ I hadn't, but then I wanted to eat ramen. (I probably don't now.)
In addition, 싶다 and most(?) other feeling-related adjectives usually take -하다 for 2nd/3rd person (except for 2nd person questions). Using 기쁘다 be happy as an example:
친구가 와서 기쁘다. I'm glad that (my) friend came.
친구가 와서 기쁘니? Are you glad that (your) friend came?
너 어제 친구가 왔다고 무척 기쁘더라. (X)
너 어제 친구가 왔다고 무척 기뻐하더라. (O) You looked so glad yesterday that your friend came.
철수가 어제 친구가 왔다고 무척 기쁘더라. (X)
철수가 어제 친구가 왔다고 무척 기뻐하더라. (O) Chulsoo looked so glad yesterday that his friend came.
철수는 어제 친구가 와서 무척 기뻤다. Chulsoo was so glad yesterday that his friend came. (This kind of form is used only in a story, where the narrator can see inside Chulsoo's mind. It can't be used if you're talking about your real friend Chulsoo.)
So, answering your question, 누가 얘기하고 싶어져요 is impossible, because (1) 누가 is third person and (2) -져요 means someone is beginning to feel that they want to talk, which is rather weird in this context.
However, you could say, e.g., 그 영화를 보고 너와 얘기하고 싶어졌어 After seeing the movie, I want to talk to you. (Or, The movie made me want to talk to you.)